Source labels openChecked July 26, 2026

Mathoverflow 347178: Bounded Only

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Does the equality $\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$ hold when both suprema are finite?

Real functions AMS 26

Mathematical statement

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Does the equality $\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$ hold when both suprema are finite?

Statement source: MathOverflow statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

mathoverflow_347178.variants.bounded_only

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem mathoverflow_347178.variants.bounded_only :    answer(sorry) ↔ ∀ᵉ (n ≥ 2) (f : ℝ^n → ℝ) (hf : ContDiff ℝ 1 f)        (h : BddAbove (range f)) (h' : BddAbove (range (fun x ↦ f (x + gradient f x)))),        (⨆ x, f x) = ⨆ x, f (x + gradient f x) := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization mathoverflow/347178background *Biagio Ricceri*background